Collective singularities of families of analytic functions

Author

Wilson, Alan

Date

1958

Degree

Doctor of Philosophy

Abstract

In his thesis Johnson gives the following definition of the radius of regularity of a family of holomorphic functions: Let F denote a family of functions f(z) regular at Z0. R is called the radius of regularity of F at z0 if R is the largest number r such that each function is holomorphic and the family is normal in |z - z 0| &lt; r. If the conditions are valid in |z - z0| &lt; r for each r > 0, then R = infinity. If the conditions are not valid in |z - z0| &lt; r for any r > 0, then R = 0. If a function of F has a singularity at z0, then R = 0.
In the present thesis we will be concerned only with the case in which F = {f(z)} = { n=0infinity afn (z - z0)n} is uniformly bounded in some neighborhood of Z0. As Johnson proves, the radius of regularity R is then given by the formula 1/R=limn&rarr; infinity supf&isin;F &vbm0;afn&vbm0;1/n where we take R = 0 whenever the righthand expression is infinite.